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arXiv · 2609.27574

Modular integrals for individual modular knots

Toshiki Matsusaka
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Abstract

To each modular knot, we attach a weight $2$ modular integral for $\mathrm{SL}_2(\mathbb{Z})$ whose homogenized cycle integrals recover its linking numbers with other modular knots in $S^3$. This construction answers Choie-Zagier's question of explicitly constructing modular integrals with prescribed rational period functions in weight $2$. Our formula applies to individual modular knots without symmetrization, thereby realizing an approach discussed but not pursued by Duke-Imamoglu-Tóth. This gives an answer to Ghys' question on pairwise linking numbers in terms of modular integrals, as sought by Simon.

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Toshiki Matsusaka. 2026-09-23. Modular integrals for individual modular knots. https://arxiv.org/abs/2609.27574

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